scholarly journals Asymptotic expansions for distributions of sums of a double sequence of quasilattice random variables

2011 ◽  
Vol 52 ◽  
pp. 353-358
Author(s):  
Algimantas Bikelis ◽  
Juozas Augutis ◽  
Kazimieras Padvelskis
Keyword(s):  
Asymptotic Expansion ◽  
Probability Distribution ◽  
Asymptotic Expansions ◽  
Probability Distributions ◽  
Double Sequence ◽  
Random Variables ◽  
Independent Random Variables ◽  
Infinitely Divisible ◽  
Formal Asymptotic Expansion

We consider the formal asymptotic expansion of probability distribution of the sums of independent random variables. The approximation was made by using infinitely divisible probability distributions.  

scholarly journals Asymptotic expansions for distribution of sums quasi-lattice random variables

2011 ◽  
Vol 52 ◽  
pp. 359-364
Author(s):  
Algimantas Bikelis ◽  
Kazimieras Padvelskis ◽  
Pranas Vaitkus
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Independent Random Variables

Althoug Chebyshev [3] and Edeworth [5] had conceived of the formal expansions for distribution of sums of independent random variables, but only in Cramer’s work [4] was laid a proper foundation of this problem. In the case when random variables are lattice Esseen get the asymptotic expansion in a new different form. Here we extend this problem for quasi-lattice random variables.  

scholarly journals Asymptotic Expansions for Distributions of Compound Sums of Random Variables with Rapidly Varying Subexponential Distribution

2007 ◽  
Vol 44 (3) ◽  
pp. 670-684 ◽  
Author(s):  
Ph. Barbe ◽  
W. P. McCormick ◽  
C. Zhang
Keyword(s):  
Asymptotic Expansion ◽  
Weibull Distribution ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Subexponential Distribution ◽  
Independent Random Variables ◽  
Sums Of Random Variables

We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same rapidly varying subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, seven-term expansions.

scholarly journals Max Weibull-G Power Series Distributions

2021 ◽  
pp. 15-29
Author(s):  
Munteanu Bogdan Gheorghe
Keyword(s):  
Probability Distribution ◽  
Power Series ◽  
Probability Distributions ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
New Family ◽  
The Family ◽  
Distribution Family ◽  
Power Series Distributions

Based on the Weibull-G Power probability distribution family, we have proposed a new family of probability distributions, named by us the Max Weibull-G power series distributions, which may be applied in order to solve some reliability problems. This implies the fact that the Max Weibull-G power series is the distribution of a random variable max (X1 ,X2 ,...XN) where X1 ,X2 ,... are Weibull-G distributed independent random variables and N is a natural random variable the distribution of which belongs to the family of power series distribution. The main characteristics and properties of this distribution are analyzed.

Infinitely divisible approximations for discrete nonlattice variables

2003 ◽  
Vol 35 (04) ◽  
pp. 982-1006
Author(s):  
V. Čekanavičius
Keyword(s):  
Asymptotic Expansions ◽  
Random Variables ◽  
Second Order ◽  
Independent Random Variables ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Compound Poisson ◽  
Set Of Points ◽  
Poisson Measures

Sums of independent random variables concentrated on discrete, not necessarily lattice, set of points are approximated by infinitely divisible distributions and signed compound Poisson measures. A version of Kolmogorov's first uniform theorem is proved. Second-order asymptotic expansions are constructed for distributions with pseudo-lattice supports.

Infinitely divisible approximations for discrete nonlattice variables

2003 ◽  
Vol 35 (4) ◽  
pp. 982-1006 ◽  
Author(s):  
V. Čekanavičius
Keyword(s):  
Asymptotic Expansions ◽  
Random Variables ◽  
Second Order ◽  
Independent Random Variables ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Compound Poisson ◽  
Set Of Points ◽  
Poisson Measures

Sums of independent random variables concentrated on discrete, not necessarily lattice, set of points are approximated by infinitely divisible distributions and signed compound Poisson measures. A version of Kolmogorov's first uniform theorem is proved. Second-order asymptotic expansions are constructed for distributions with pseudo-lattice supports.

scholarly journals Asymptotic Expansions for Distributions of Compound Sums of Random Variables with Rapidly Varying Subexponential Distribution

2007 ◽  
Vol 44 (03) ◽  
pp. 670-684 ◽  
Author(s):  
Ph. Barbe ◽  
W. P. McCormick ◽  
C. Zhang
Keyword(s):  
Asymptotic Expansion ◽  
Weibull Distribution ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Subexponential Distribution ◽  
Independent Random Variables ◽  
Sums Of Random Variables

We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same rapidly varying subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, seven-term expansions.

A note on passage times and infinitely divisible distributions

1967 ◽  
Vol 4 (2) ◽  
pp. 402-405 ◽  
Author(s):  
H. D. Miller
Keyword(s):  
Random Walk ◽  
Markov Process ◽  
Continuous Time ◽  
Random Variables ◽  
Independent Random Variables ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Time Intervals ◽  
Mutually Independent ◽  
Passage Times

Let X(t) be the position at time t of a particle undergoing a simple symmetrical random walk in continuous time, i.e. the particle starts at the origin at time t = 0 and at times T1, T1 + T2, … it undergoes jumps ξ1, ξ2, …, where the time intervals T1, T2, … between successive jumps are mutually independent random variables each following the exponential density e–t while the jumps, which are independent of the τi, are mutually independent random variables with the distribution . The process X(t) is clearly a Markov process whose state space is the set of all integers.

scholarly journals Uniform renewal theory with applications to expansions of random geometric sums

2007 ◽  
Vol 39 (4) ◽  
pp. 1070-1097 ◽  
Author(s):  
J. Blanchet ◽  
P. Glynn
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Renewal Theory ◽  
Random Variable ◽  
Higher Order ◽  
Order Correction ◽  
Diffusion Approximations ◽  
Geometric Sums ◽  
Correction Terms

Consider a sequence X = (Xn: n ≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variable M with parameter p. The random variable SM = X1 + ∙ ∙ ∙ + XM is called a geometric sum. In this paper we obtain asymptotic expansions for the distribution of SM as p ↘ 0. If EX1 > 0, the asymptotic expansion is developed in powers of p and it provides higher-order correction terms to Renyi's theorem, which states that P(pSM > x) ≈ exp(-x/EX1). Conversely, if EX1 = 0 then the expansion is given in powers of √p. We apply the results to obtain corrected diffusion approximations for the M/G/1 queue. These expansions follow in a unified way as a consequence of new uniform renewal theory results that are also developed in this paper.

Introduction to Information Theory

2014 ◽  
Author(s):  
M. Vidyasagar
Keyword(s):  
Information Theory ◽  
Probability Distribution ◽  
Relative Entropy ◽  
Probability Distributions ◽  
Random Variables ◽  
Conditional Entropy ◽  
Random Variable ◽  
Entropy Function ◽  
Concave Functions ◽  
Leibler Divergence

This chapter provides an introduction to some elementary aspects of information theory, including entropy in its various forms. Entropy refers to the level of uncertainty associated with a random variable (or more precisely, the probability distribution of the random variable). When there are two or more random variables, it is worthwhile to study the conditional entropy of one random variable with respect to another. The last concept is relative entropy, also known as the Kullback–Leibler divergence, which measures the “disparity” between two probability distributions. The chapter first considers convex and concave functions before discussing the properties of the entropy function, conditional entropy, uniqueness of the entropy function, and the Kullback–Leibler divergence.

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